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58
Modeling Spatial Relations and Operations with Partially Ordered Sets
 INTERNATIONAL JOURNAL OF GEOGRAPHICAL INFORMATION SYSTEMS
, 1993
"... Formal methods for the description of spatial relations can be based on mathematical theories of order. Subdivisions of land are represented as partially ordered sets (posets), a model that is general enough to answer spatial queries about inclusion and containment of spatial areas. After a brief in ..."
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Cited by 27 (4 self)
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Formal methods for the description of spatial relations can be based on mathematical theories of order. Subdivisions of land are represented as partially ordered sets (posets), a model that is general enough to answer spatial queries about inclusion and containment of spatial areas. After a brief introduction to the basic concepts of posets and lattices, their applications to modeling spatial relations and operations for spatial regions in terms of containment and overlay are presented. An interpretation is given for new geographic elements that are created by the completion from a poset to a lattice. It is shown that a novel approach to characterize certain topological relations based on a lattice of a simplicial complex is a model for spatial regions that combines both topological and order relations and allows spatial queries to be answered in a unified way.
Minimal surfaces extend shortest path segmentation methods to 3D
 IEEE Transactions on PAMI
, 2010
"... Abstract—Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. Howeve ..."
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Cited by 24 (2 self)
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Abstract—Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. However, many different extensions of the shortest path techniques to 3D have been previously proposed in which the 3D object is segmented via a collection of shortest paths rather than a minimal surface, leading to a solution which bears an uncertain relationship to the true minimal surface. Specifically, there is no guarantee that a minimal path between points on two closed contours will lie on the minimal surface joining these contours. We observe that an elegant solution to the computation of a minimal surface on a cellular complex (e.g., a 3D lattice) was given by Sullivan [47]. Sullivan showed that the discrete minimal surface connecting one or more closed contours may be found efficiently by solving a Minimumcost Circulation Network Flow (MCNF) problem. In this work, we detail why a minimal surface properly extends a shortest path (in the context of a boundary) to three dimensions, present Sullivan’s solution to this minimal surface problem via an MCNF calculation, and demonstrate the use of these minimal surfaces on the segmentation of image data. Index Terms—3D image segmentation, minimal surfaces, shortest paths, Dijkstra’s algorithm, boundary operator, total unimodularity, linear programming, minimumcost circulation network flow. Ç 1
How to write mathematics
 L'Enseignment Mathematique
, 1970
"... This is a subjective essay, and its title is misleading; a more honest title might be how i write mathematics. It started with a committee of the American Mathematical Society, on which I served for a brief time, but it quickly became a private project that ran away with me. In an effort to bring it ..."
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Cited by 18 (0 self)
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This is a subjective essay, and its title is misleading; a more honest title might be how i write mathematics. It started with a committee of the American Mathematical Society, on which I served for a brief time, but it quickly became a private project that ran away with me. In an effort to bring it under control I asked a few friends to read it and criticize it. The criticisms were excellent; they were sharp, honest, and constructive; and they were contradictory. "Not enough concrète examples " said one; "don't agrée that more concrète examples are needed " said another. "Too long" said one; "maybe more is needed " said another. "There are traditional (and effective) methods of minimizing the tediousness of long proofs, such as breaking them up in a séries of lemmas " said one. "One of the things that irritâtes me greatly is the custom (especially of beginners) to présent a proof as a long séries of elaborately stated, utterly boring lemmas" said another. There was one thing that most of my advisors agreed on; the writing
Topologically Defined Isosurfaces
 IN PROC. 6TH DISCRETE GEOMETRY FOR COMPUTER IMAGERY
, 1996
"... In this research report, we present a new process for defining and building the set of configurations of MarchingCubes algorithms. Our aim is to extract a topologically correct isosurface from a volumetric image. Our approach exploits the underlying discrete topology of voxels and especially the ..."
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Cited by 14 (0 self)
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In this research report, we present a new process for defining and building the set of configurations of MarchingCubes algorithms. Our aim is to extract a topologically correct isosurface from a volumetric image. Our approach exploits the underlying discrete topology of voxels and especially their connectedness. Our main
A convenient differential category
, 2011
"... We show that the category of convenient vector spaces in the sense of Frölicher ..."
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Cited by 13 (2 self)
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We show that the category of convenient vector spaces in the sense of Frölicher
Morse theory for filtrations and efficient computation of persistent homology, in preparation
"... Abstract. We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations. ..."
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Cited by 10 (2 self)
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Abstract. We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.
The Shuffle Hopf Algebra and Noncommutative Full Completeness
, 1999
"... We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformati ..."
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Cited by 8 (3 self)
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We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cutfree proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *autonomous category, canonically enriched over vector spaces. This paper
The Lefschetz coincidence theory for maps between spaces of different dimensions
, 2000
"... For a given pair of maps f, g: X → M from an arbitrary topological space to an nmanifold, the Lefschetz homomorphism is a certain graded homomorphism Λfg: H(X) → H(M) of degree (−n). We prove a Lefschetztype coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x ∈ X ..."
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Cited by 6 (5 self)
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For a given pair of maps f, g: X → M from an arbitrary topological space to an nmanifold, the Lefschetz homomorphism is a certain graded homomorphism Λfg: H(X) → H(M) of degree (−n). We prove a Lefschetztype coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x ∈ X such that f(x) = g(x).